Acknowledgments

characteristic can be prescribed to allow linear operation along with near classical efficiency. Section 1.4 discusses the RF bipolar and its radically different formulation for reaching the same goal of linearity combined with highefficiency operation. The RF bipolar emerges from this analysis, taking full account of the discussion in Section 1.3, in a surprisingly favorable light. Section 1.5 returns to the field effect transistor (FET) as a Class AB device, and the extent to which existing devices can fortuitously exhibit some of the linearization possibilities discussed in Section 1.3.

1.2 Classical Class AB Modes

This analysis should need no introduction, and what follows is largely a summary of a more detailed treatment in RFPA, but with some extensions into the possibilities offered by dynamically varying RF loads. Figure 1.1 shows an idealized RF device, having a linear transconductive region terminated by a sharply defined cutoff point. The device is assumed to be entirely transconductive, that is to say, the output current has no dependency on the output voltage provided this voltage is maintained above the turn-on, or “knee” value, Vk. The analysis will further make the approximation that Vk is

Saturation region

Imax

Drain current

Quasi-linear region

Imax /2

Cutoff region

0

Gate voltage (normalized)

Figure 1.1 Ideal transconductive device transfer characteristic.

negligible in comparison to the dc supply voltage, in other words, zero. This approximation is conspicuously unreal, and needs immediate addressing if the voltage is anything other than sinusoidal, but is commonplace in elementary textbooks. Figure 1.2 shows the classical circuit schematic for Class AB operation. The device is biased to a quiescent point which is somewhere in the region between the cutoff point and the Class A bias point. The input drive level is adjusted so that the current swings between zero and Imax, Imax

Vdc

iD Imax

Drain current

Idc

0

Vo

vDS

Drain voltage

Figure 1.2 Class AB amplifier: schematic and waveforms.

being a predetermined maximum useable current, based on saturation or thermal restrictions.

The resulting current waveforms take the form of asymmetrically truncated sinewaves, the zero current region corresponding to the swings of input voltage below the cutoff point. These current waveforms clearly have high harmonic content. The key circuit element in a Class AB amplifier is the harmonic short placed across the device which prevents any harmonic voltage from being generated at the output. Such a circuit element could be realized, as shown in Figure 1.2, using a parallel shunt resonator having a resonant frequency at the fundamental. In principle the capacitor could have an arbitrarily high value, sufficient to short out all harmonic current whilst allowing the fundamental component only to flow into the resistive load. So the final output voltage will approximate to a sinewave whose amplitude will be a function of the drive level and the chosen value of the load resistor. In practice the load resistor value will be chosen such that at the maximum anticipated drive level, the voltage swing will use the full available range, approximated in this case to an amplitude equal to the dc supply. For the purposes of this analysis, the maximum drive level will be assumed to be that level which causes a peak current of Imax.

Some simple Fourier analysis [1] shows that the efficiency, defined here as the RF output divided by the dc supply, increases sharply as the quiescent bias level is reduced, and the so-called conduction angle drops (Figure 1.3). Not only does this apply to the efficiency at the designated maximum drive level, but the efficiency in the “backed-off” drive condition also increases, especially in relation to the Class A values (Figure 1.4). What is less familiar is the plot of linearity in the Class AB region, shown in Figure 1.5. The process of sharp truncation of the input sinusoidal signal unfortunately generates some less desirable effects; odd degree distortion is part of the process and gain compression is clearly visible anywhere in the Class AB region. This gain compression comes from a different, and additional, source than the gain compression encountered when a Class A amplifier, for example, is driven into saturation. Saturation effects are primarily caused by the clipping of the RF voltage on the supply rails. The class AB nonlinearity in Figure 1.5 represents an additional cause of distortion which will be evident at drive levels much lower than those required to cause voltage clipping. This form of distortion is particularly undesirable in RF communications applications, where signals have amplitude modulation and stringent specifications on spectral spreading.

The Class B condition, corresponding to a zero level of quiescent bias, is worthy of special comment. This case corresponds to a current waveform

RF power (dB)

0

RF power (dB)

Figure 1.3 Reduced conduction angle modes, power and efficiency at maximum drive level.

100

Efficiency (%)

Iq = 0 (Class B)

50

Figure 1.4 Efficiency as a function of input drive backoff (PBO) and Class AB “quiescent” current (Iq) setting.

Vq = 0 (Class B)

Input power (2 dB/div)

Figure 1.5 Class AB gain characteristics.

which, within the current set of idealizing assumptions, is a perfectly halfwave rectified sinewave. Such a waveform contains only even harmonics, and in the absence of damaging odd degree effects, the backed-off response in Figure 1.5 shows a return to linear amplification. In practice, such a desirable situation is substantially spoiled by the quirky, or at best unpredictable, behavior of a given device so close to its cutoff point. It is frequently found, usually empirically, that a bias point can be located some way short of the cutoff point where linearity and efficiency have a quite well-defined optimum. Such “sweet spots” are part of the folklore of RFPA design, and some aspects of this subject will be discussed in more detail in Section 1.3.

One additional aspect of Class AB operation which requires further consideration is the issue of drive level and power gain. It is clear from Figure 1.2 that as the quiescent bias point is moved further towards the cutoff point, a correspondingly higher drive voltage is required in order to maintain a peak current of Imax. In many cases, especially in higher RF or microwave applications, the gain from a PA output stage is a hard-earned and critical element in the overall system efficiency and cost. In moving the bias point from the Class A (Imax/2) point to the Class B (zero bias) point, an increase of drive

are firstly the rapid drop in efficiency as a modulated signal drops to low envelope amplitudes, and secondly the need to control power over a wide dynamic range [for example, in code division multiple access (CDMA) systems], this configuration appears to fulfill both goals handsomely. There are also, of course, two immediate problems; the device is a nonlinear amplifier having a square-root characteristic, and we have so far ignored the practical issue of how such a dynamic load variation could be realized in practice.

As will be discussed in Chapter 2, the realization of an RF power amplifying system capable of performing the feat of linear high efficiency amplification over a wide dynamic signal range has been something of a “Holy Grail” of RFPA research for over half a century. Both the Doherty and Chireix techniques (Chapter 2) are candidates, but also generate a collection of additional, mainly negative, side issues. The fundamental principle remains sound, and is an intriguing goal for further innovative research.

1.3 Class AB: A Different Perspective

The idealized analysis of Class AB modes summarized in Section 1.2 raises a number of issues for those who have experience in using such amplifiers in practice. Most prominently, the assumption of a linear transconductive device is an idealization that is unsatisfactory for just about any variety of RF device in current use, whether it be an FET or bipolar junction transistor (BJT). It seems that in practice the use of an imperfect device can fortuitously reduce the nonlinearities caused by the use of reduced angle operation. This section explores this extension to the theory and comes up with some proposals concerning the manner in which RF power transistors should be designed and specified. In essence, devices with substantial, but correctly orientated, nonlinear characteristics are required to make power amplifiers having the best tradeoff between efficiency and linearity. The process of defining such devices involves some basic mathematical analysis and flagrantly ignores, for the present purposes, the technological issues involved in putting the results into practice. This is a necessary and informative starting point.

The analysis in Section 1.2 showed that an ideal transconductive device, biased precisely at its cutoff point, gives an optimum linear amplifier, having high efficiency and a characteristic which contains even, but not odd, degree nonlinearity. This is the classical Class B amplifier. It has already been commented that in practice, true “zero-bias” operation usually yields unsatisfactory performance, especially at well backed-off drive levels where the device will typically display a collapse of small signal gain. Even a device with

Figure 1.8 Square-law and cube-law device characteristics, compared to ideal device using linear and cutoff regions. (Note changed normalization for ideal device.)

and for maximum current swing under sinusoidal excitation, the quiescent bias point will be set to vi = 0.5, and the input signal will be to vi = vs cosq, with vs varying between zero and a maximum value of 0.5. So the output current for this device will be given by

Assuming that the output matching network presents a short circuit at all harmonics of q, the fundamental output voltage amplitude is a linear function of the input level, vs, despite the square-law device characteristic.

Compared to a device with an ideal linear characteristic in Class A operation, where

io = (21 + v s cos q)

the square-law device has the same fundamental output amplitude, but a dc component reduced by a factor of

which improves the output efficiency over the corresponding linear Class A value. So at the maximum drive level of vs = 0.5, the efficiency of the squarelaw device is 2/3 or 66.7%. This improvement in efficiency is obtained simultaneously with perfectly linear amplification.

A cube-law device (Figure 1.8), on the other hand, gives substantial improvement in efficiency at the expense of linearity,

(1.3)

showing an increased fundamental component compared to the linear case, and a reduced dc component. The output efficiency,

is now 3/4, or 75%, at maximum current swing (vs = 0.5), but at this drive level the device displays 1.9 dB of gain expansion, leading to substantial generation of third-degree nonlinearities.

It is therefore apparent that to create a device which has optimum efficiency and perfect linearity, it is necessary to tailor the transfer characteristic to generate only even powers of the cosine input signal. Unfortunately, this is not as simple as creating a power transfer characteristic having a higher even order power,

io = (21 + v s cos q)4

which will contain both even and odd powers of the cosine signal.

The necessary characteristic can be determined by finding suitable coefficients of the even harmonic series,

io = ko + cos q + k2 cos 2q + k4 cos4q + k6 cos6q + K + k2n cos 2nq

normalized such that 0 < i0 < 1.

The goal here is to find a set of coefficients which generates a waveform having the same peak-to-peak swing, from zero to unity, but which has decreasing mean value as successive even harmonic components are added. The optimum case will be a situation where the negative half cycles have a maximally flat characteristic at their minima; this corresponds to values of the kn coefficients determined by setting successive derivatives of the function

f (q) = cos q + k2 cos 2q + k4 cos4q + k6 cos6q + K + k2n cos 2nq

equal to zero at q = p. This generates a set of simultaneous equations for the kn values. Some of the resulting waveforms are shown in Figure 1.9, from which it is clear that as more even harmonics are added, the resulting waveforms more closely approach an ideal Class B form. Table 1.1 shows the values for a number of values of n, along with the efficiency, which improves for higher values of n due to the decreasing values of mean current, k0.

Clearly, Table 1.1 shows that a useful increase in efficiency can be obtained for a few values of n above the simple square-law case of n = 1. Conversely, the large n values required to approach the Class B condition are unlikely to be realized in practice due to the limited switching speed of a typical RF transistor.

1

n = 1, 2, 4, 12

io

q

Figure 1.9 Current waveforms having “maximally flat” even harmonic components (n factor indicates the number of even harmonics).

It is a simple matter to convert the desired current waveforms shown in Figure 1.9 into corresponding transfer characteristics, assuming a sinusoidal voltage drive. These corresponding nonlinear transconductances are shown in Figure 1.10. It seems that an unfamiliar device characteristic emerges from this simple analysis, which displays efficiency in the mid-70% region and has only even order nonlinearities. It has a much slower turn-on characteristic than the classical FET dogleg, and resembles a bipolar junction transistor (BJT), rather than an FET in its general appearance. Figure 1.10 also shows that the desired family of linear, highly efficient characteristics fall into a well-defined zone. The boundaries of the zone are formed by the square-law characteristic, and the classical Class B dogleg. It is interesting to plot some other characteristics on the same chart, as shown in Figure 1.11. The characteristics which have inherent odd degree nonlinearities always cross over the boundary formed by the dogleg. The linearity zone, thus defined, would appear to be a viable and realistic target for device development.

The chart of Figure 1.11 has some interesting implications for the future of RF power bipolars. This will be further discussed in Section 1.4. FETs, however, do not fare so well in this analysis. An FET will usually display a closer approximation to a dogleg characteristic; this is a natural outcome of their modus operandi, coupled with some misdirected beliefs on the part of manufacturers as to what constitutes a “good” device characteristic. It could be reasonably argued that a typical FET characteristic has the appearance of one of the higher n-value curves in Figure 1.10, having linear transconductance with a short turn-on region. Such a device would, within the idealized boundaries of the present analysis, still comply with the

requirements of even degree nonlinearity and higher efficiency than the lower n-value curves. The problem with this kind of device lies in the precision of the quiescent bias setting, which leads to more general issues of processing yield. It is fair to speculate that the unfamiliar-looking n = 4 curve, for example, would be a more robust and reproducible device for linear power applications.

1.4 RF Bipolars: Vive La Difference

The idealized analysis of Class AB modes summarized in Section 1.1 has its roots in tube amplifier analysis and dates from the early part of the last century. The early era of RF semiconductors was dominated by a radically different kind of device, the bipolar transistor. More recently, the emergence of RF FET technologies, such as the Gallium Arsenide Metal Semiconductor Field Effect Transistor (GaAs MESFET) and Silicon Metal Oxide Semiconductor (Si MOS) transistor, has renewed the relevance of the older traditional analysis. Strangely, it seems that despite some obvious and fundamental physical differences in the manner of operation of BJTs, much of the conceptual framework and terminology of the traditional analysis seems to have been retained by the BJT RFPA community. This has required the application of some hand-waving arguments which seek to gloss over the major physical differences that still exist between BJT and FET device operation.

This section attempts to perform a complementary analysis of a BJT RF power amplifier, in the same spirit of device model simplicity as was used in analyzing the FET PAs in Section 1.1. Unfortunately, the exponential forward transfer characteristic of the BJT device will necessitate greater use of numerical, rather than purely analytical, methods. It will become clear that the BJT is a prime candidate for practical, and indeed often fortuitous, implementation of some of the theoretical results discussed in this section.

1.4.1A Basic RF BJT Model

The model for the RF BJT which will be used is shown in Figure 1.12. This model incorporates the two essential textbook features of BJT operation: a base-emitter junction which has an exponential diode I-V characteristic, and a collector-emitter output current generator which supplies a multiplied replica of the current flowing in the base-emitter junction. As with the FET model, all parasitic elements are assumed to be either low enough to be ignored or to form part of the external matching networks which resonate

Imax, which will usually be based on thermal considerations for a BJT. This maximum current will be normalized to unity in the following analysis. There is an additional issue in the normalization process for a BJT, which is the steepness of the exponential base-emitter characteristic. For convenience, this will be modeled using values which give a typical p-n junction characteristic which turns on over approximately 10% of a normalized vb range of 0 to 1. So

where a k value of 7 and a normalized b value of unity give the characteristics shown in Figure 1.13. This closely resembles a typical BJT device except that the “on” voltage, where ic = 1, is normalized to unity.1

Clearly, the immediate impression from Figure 1.13 is of a highly nonlinear device. But this impression can be tempered by the realization that, unlike in the previous FET analysis, the voltage appearing across the baseemitter junction is no longer a linear mapping of the voltage appearing at the terminals of the RF generator; the input impedance of the RF BJT also displays highly nonlinear characteristics. It is the interaction of these two nonlinear effects which has to be unraveled, to gain a clear understanding of how one can possibly make linear RFPAs using such a device.

As usual, our elementary textbooks from younger days have a simple solution, and there is a tendency for this concept to be stretched, in later life, well beyond its original range of intended validity. Basically, if the device is fed from a voltage generator whose impedance, either internal to the generator or through the use of external circuit elements, is made sufficiently high compared to the junction resistance, then the base-emitter current approximates to a linear function of the generator voltage, which in turn appears in amplified form in the collector-emitter output circuit. This process is illustrated in Figure 1.14, where the effect of placing a series resistor on the base is shown for a wide range of normalized resistance values. The curves in Figure 1.14 are obtained by numerical solution of the equation

where R is normalized to 1W for normalized unity values of current and voltage.

1.It is also convenient to normalize b to unity, so that ib and ic are both normalized over a range of 0 to 1.

This simplification of BJT operation is the mainstay of most lowfrequency analog BJT circuit design, but it has two important flaws in RFPA applications. The first problem concerns the optimum use of the available generator power. In RF power applications, power gain is usually precious and the device needs to be matched close to the point of maximum generator power utilization. This will typically imply a series resistance that is much lower in value than that required to realize the more extensive linearization effects shown in Figure 1.14. The second problem is that in order to make a Class AB type of amplifier, the output current waveform, and correspondingly the base-emitter current, has to be highly nonlinear. Thus the input series resistor has to be a “real” resistor, having linear broadband characteristics. This will not be the case if the series resistance is realized using conventional matching networks at the fundamental frequency. The fact that in a BJT the collector and base currents have to maintain a constant linear relationship is a crucial difference between FET and BJT amplifiers running in Class AB modes, and leads directly to the inconvenient prospect of harmonic impedance matching on the input, as well as the output.

Considering the Class A type operation initially, the need for a high Q input resonant matching network does enable the low frequency “currentgain” concept to be stretched into use. Figure 1.15 shows a situation closer to

R = 1, 2, 5

R = 0.5

R = 0

0

Vin /V1

Figure 1.14 BJT transfer characteristics for varying base resistance. (Voltage scale normalized to V1, the value of Vin required for ib = 1 at each selected R value.)

Major problems will be encountered, however, if attempts are made to run this circuit configuration in a Class AB mode. Any effort to force a nonsinusoidal current in the base-emitter junction (such as by reducing the quiescent bias voltage) conflicts with the resonant properties of the input matching network, which will strongly reject harmonics through its high above-resonance impedance. In practice, the resonance of the input matching network may have only a moderate Q factor, and will allow some higher harmonic components to flow, giving some rather quirky approximations to Class AB or B operation. The harmonic current flow may also be aided by the BJT base-emitter junction capacitance, which will form part of the input-matching resonator. As discussed in RFPA, in connection with output harmonic shorts (see pp. 108–110), this leads to a curious irony in that higher frequency devices with lower parasitics can be harder to use at a given frequency from the harmonic trapping viewpoint. At 2 GHz, a typical Si BJT device will have an input which is dominated by a large junction capacitance. Although this makes the fundamental match a challenging design problem, it does have an upside in that higher harmonics will be effectively shorted out. A 40-GHz heterojunction bipolar transistor (HBT) device, however, will need assistance in the form of external harmonic circuitry.

Returning to the transfer characteristics plotted in Figure 1.14, it should be apparent that the intermediate values for series resistance give curves which are quite similar to those generated speculatively in Section 1.2 (see Figure 1.11). The BJT device appears to be a ready-made example of the novel principle that Class AB PAs can be more linear if the device has the right kind of nonlinearity in its transfer characteristic. This can be explored in a more quantitative fashion by taking the transfer characteristics in Figure 1.14 and subjecting the device to sinusoidal excitation. Figure 1.17(a) shows the circuit and defines the excitation. For convenience, the dc bias is assumed to be applied at the RF generator, and for the time being the input resistor is assumed to be a physical resistor, encompassing both the matched generator impedance and any additional resistance on the base. It should be noted that such a circuit configuration assumes that the resistor is a true resistor at all relevant harmonic frequencies. Although the input matching network can be assumed to transform the generator impedance to the R value at the fundamental, the circulating harmonic components of base current need to be presented with the same resistance value. One possible more practical configuration for realizing this requirement is shown in Figure 1.17(b). This initial analysis returns to the original assumption of ignoring the base-emitter capacitance; this approximation will be reviewed at a later stage.

Figure 1.17 BJT Class AB circuits: (a) schematic for analysis; and “broadband” components and (b) possible practical implementation.

Unfortunately, the exponential base-emitter characteristic defies an analytical solution for the current flowing in the circuit of Figure 1.17(a), when using the instantaneous generator voltage as the independent input variable. An iteration routine has to be used at each point in the RF cycle to determine the junction current. A typical set of resulting waveforms is shown in Figure 1.18. These waveforms show three different cases of dc bias, resulting in corresponding quiescent current (Iq) values, for an input voltage swing chosen to give a stipulated maximum peak current (Imax) for the device. These current waveforms clearly resemble classical Class AB form, but are not precisely the same and have a complicated functional relationship with the

0

t

Figure 1.18 BJT “Class AB” current waveforms.

selected series resistance (input match), the bias point (which at this time is incorporated into the RF drive and has the same series resistance), and the RF drive level. Using these three essentially independent variables, we can explore the relationship between efficiency and linearity, at comparative output power levels.

The new variable factor in such an analysis is the choice of input resistance. Unlike the ideal FET analysis in Section 1.2, it now appears that the linearity of the amplifier, as well as its power gain, will have some important dependency on the selection of this circuit element. As always, it can be expected that some tradeoffs will be necessary. Power gain, efficiency, and linearity will all have different optimum values of input resistance. Figure 1.19 shows the gain compression and efficiency as a function of power backoff (PBO) for the three quiescent current settings in Figure 1.18. It is immediately clear that one case, corresponding to the “deep Class AB” quiescent current of 0.069 (6.9%), appears to give very linear power gain right up to the maximum peak current drive level, with a corresponding peak efficiency of just under 80%.

This desirable set of characteristics is closely coupled with the initial choice of normalized series resistance, R = 0.5. This value was selected on a qualitative comparison between the BJT transfer characteristics plotted in Figure 1.14 and the “linear zone” requirements suggested in Figure 1.11. It turns out that the only downside of this selection is that it represents a value substantially higher than the generator resistance required to achieve maximum power transfer to the device. Figure 1.20 shows a similar plot, but

showing three different values of series resistance (R = 0.25, 0.5, 0.75). In each case the quiescent bias point can be adjusted to give a linearity defined to be less than 0.5-dB gain variation over the 20-dB sweep range. The lower value of R = 0.25 shows a better input power match to the device, but requires a higher Iq setting in order to maintain the stipulated linearity; this lowers the efficiency. Higher R values result in similar good linearity at lower Iq settings and higher efficiency, but lower power gain. The results for R = 0.5 seem to represent a good overall compromise.

These results appear to show the BJT as a very promising device for high efficiency linear RFPA applications. The ability to use a circuit element to perform the linearization function on the transfer characteristic, as discussed in Section 1.3, is an asset which would not be available for a true transconductive device such as an FET. There is an important caveat on the above analysis, however. The junction has been assumed to be entirely resistive, albeit highly nonlinear. In practice the junction of an RF BJT will be shunted by a capacitance. Depending on the relationship between the frequency of use and the maximum frequency of the device, the impact of the junction capacitance on this analysis could be substantial. The situation is analogous to the discussion presented in RFPA (Chapter 5) concerning the effect of output capacitance of an RF power device in Class AB applications. In modern wireless communications, it is not uncommon to use a much higher frequency technology for applications below 1 GHz. A designer of HBT handset PAs using a 30-GHz HBT or pseudomorphic high electron mobility transistor (PHEMT) process could well find that the junction capacitance tends towards the low-impact extreme; a 2-GHz high power Si BJT, on the other hand, will almost certainly present a base-emitter impedance that is difficult to distinguish from a capacitor.

It is necessary, therefore, to repeat the above analysis for the opposite extreme case, where the device junction capacitance is sufficiently large that it can be assumed to act as a bypass capacitor for all harmonic current components, and is resonated out at the fundamental by the input matching network. The modified schematic diagram is shown in Figure 1.21. It is assumed now that the input-matching network effectively places a shunt resonator across the junction, such that the voltage across the junction is forced always to be sinusoidal. The input-matching network will also transform the generator impedance from its nominal 50-W value, down to form the input resistance R; R may also include some parasitic on-chip resistance. This circuit can be analyzed more easily. The input sinusoidal generator amplitude Vin will cause a corresponding change in the sinusoidal amplitude Vs appearing across the junction. Thus, if Vs is used as the chosen input signal

Vdc

RF “choke”

Figure 1.21 Schematic of BJT Class AB RFPA; junction capacitor shorts harmonic current components, but is resonated at fundamental by input matching network.

amplitude variable, it is possible to determine the current flowing in the junction,

where vq is the dc voltage bias.

This can be integrated over a cycle in order to extract the fundamental component, Ib1.

The generator voltage amplitude required, then, to satisfy the defined condition is

Vin = RI b 1 +V s

Figure 1.22 shows a comparable plot to Figure 1.18, showing a significantly modified set of waveforms for the same quiescent bias settings. The power sweep plot in Figure 1.23 can also be directly compared to the ideal junction plots shown in Figure 1.20. For a resistance setting of R = 0.5, it is clear that a higher quiescent current is required in order to achieve comparable linearity in the “harmonic short junction” case. This results in lower efficiency. However, a higher value of R = 1, shown in Figure 1.24, offers a somewhat better tradeoff, showing a peak efficiency just under 70% for comparable linearity and power gain. It is also significant that the more peaked waveforms in this case result in about 1-dB lower fundamental power than the comparable ideal junction analysis. Although the high-capacitance device shows less than the stellar performance of the ideal device, a practical device could be expected to give results somewhere between these two extreme cases.

Figure 1.24 Gain and efficiency plots, harmonic shorted junction, higher (R = 1) series resistance value.

can make linear, highly efficient PAs, but they appear to have design flexibility which makes them arguably superior to the more ubiquitous FET devices. But in order to harness this potential, several basic design issues must be observed, namely:

∙The input matching configuration, including the bias circuit, has a major impact on the operation of a BJT RFPA.

∙The input match will show different optima for maximum gain, best linearity, and highest efficiency. Optimization of the second two may involve substantial reduction in power gain.

∙Correct handling of harmonics is a necessary feature on the input, as well as the output, match. Situations where a device is being used well below its cutoff frequency may require, or will greatly benefit from, specific harmonic terminating circuit elements on the input.

∙The process of linearizing the response of a BJT includes the use of a specific, and very low, impedance for the base bias supply voltage. This is a very different bias design issue in comparison to the simple current bias used in small signal BJT amplifiers, or the simple high impedance voltage bias used in FET PAs.

∙The use of on-chip resistors in order to improve the linearity of a BJT RFPA device, as opposed to thermal ballasting, seems worthy of more extensive simulation and development efforts.

1.5On Sweet Spots and IM Glitches

Sections 1.3 and 1.4 have presented a somewhat radical approach to the design of Class AB RFPAs. Essentially, the possibility has been demonstrated to “prescribe” an ideal device characteristic which will display the efficiency advantages of conventional Class AB modes, but have greatly suppressed odd-degree distortion. In Section 1.4, it was shown that some interesting possibilities exist for implementing this approach through the use of external circuit elements. None of this, however, is of much immediate help to a designer using the FET device technologies which dominate PA design above 1 GHz. Device technologists will not typically be able to respond quickly to requests for draconian changes in their device characteristics.

On the other hand, it is a matter of common experience that some FET device types show helpful “glitches” or “suckouts” in their intermodulation (IM) or adjacent channel power (ACP) responses. The origin of this behavior can be traced along much the same lines that were followed in Section 1.3; nonlinearities in the transfer characteristic can fortuitously cancel the nonlinearities which are fundamental to Class AB operation. In many practical cases, the quiescent current setting for a particular device will be largely determined such that an IM “notch” is placed strategically near the maximum peak envelope power (PEP) drive level, so that the efficiency specification can be met.

Figure 1.25 shows a simple example of a sharp turn-on FET characteristic which has an additional gain expansion component in its “linear” region. The transfer characteristics of this device are

showing a simple third-degree gain expansion term. The expression is normalized through the parameter gz, which determines the amount of gain expansion but maintains the current such that at vin = 1, id = Imax.

Such a device, operating at an appropriately selected Class AB quiescent bias setting, will show a sharp null in its IM characteristics, as shown in Figure 1.26. Note that even outside of the null, the IM3 is still substantially reduced in comparison to the linear (gz = 1) case. Essentially, the gain

Figure 1.25 FET characteristic with gain expansion gz parameter (see text) 1, 0.85, 0.75, 0.65.

compression caused by the truncation of the current waveforms in Class AB operation will be cancelled by the gain expansion built into the device characteristic. A simple third-degree expander such as this can only cancel or reduce third-degree effects. In practice, any device having gain expansion will have higher-degree components, and multiple nulls in higher order IMs are often observed. Figure 1.26 also shows a couple of apparent downsides; the power gain is reduced, surprisingly, as the gain expansion factor is increased. This is due to the reduction in gain at low drive levels and is actually more an artifact of the normalization of Imax than being a fundamental tradeoff. Figure 1.26 also shows the ideal linear device as having no IM or distortion at drive levels lower than the onset of Class AB truncation. This simply is a result of using an ideal linear transconductance (gz = 1).

It is important to recognize that these effects may not in practice be simply attributed to the nonlinearity of the transfer characteristics. Nonlinearity in the device parasitics, especially the junction capacitances, can also play a part in creating sufficient nonlinearity in the device gain characteristics to provide cancellation of the Class AB compression at a specific power level. Such nulling phenomena can be difficult to preserve on a wafer-to-wafer basis, and over an extended period. But these effects demonstrate the general principle that device characteristics can be tailored to increase the linearity of efficient Class AB amplifiers.

harness the value of even degree distortion. A more logical approach in the modern era, where semiconductor wafers can be prescribed in three dimensions with close to molecular precision, is to synthesize device characteristics which retain the advantages of the traditional approach but reduce the disadvantages. Conventional Class AB operation incurs odd degree nonlinearities in the process of improving efficiency. Mathematical reasoning shows that it is feasible to specify a device characteristic that increases efficiency all the way up to 78% by the use of only even order nonlinearities. Such a device will not generate undesirable close-to-carrier intermodulation distortion.

The bipolar device emerges from this analysis very favorably, so long as its parasitics can be minimized at the frequency of intended use. The pragmatist may well find much to question in the analyses and device models used in this chapter. Parasitic reactances, especially nonlinear ones, will affect the waveforms and the conclusions. But as PA device technology advances, it is relevant and appropriate to idealize device behavior. At lower frequencies in the RF spectrum devices having cutoff frequencies in the 30–50-GHz region can show performance quite close to ideal.

Reference

[1]Cripps, S. C., RF Power Amplifiers for Wireless Communications, Norwood, MA: Artech House, 1999.

The technique has an Achilles’ heel—the need to convert a suitably profiled, linearized PA supply drive signal to the necessary high level of current and voltage required by the PA itself. This not only erodes the efficiency advantage, through the additional efficiency factor of a power converter, but also has an important impact on the maximum viable signal bandwidth. There is also a secondary issue of dynamic range; most RFPAs of standard design will display sufficient transmission, even with zero supply, to limit the dynamic range of the system to about 20 dB.

Successful implementations of EER which demonstrate some alleviation to these problems have appeared in recent literature [2]. The focus in this chapter, however, is to follow some alternative avenues in pursuit of a PA design technique which conserves efficiency in wide dynamic signal range applications; the methods proposed in two classical papers from the 1930s, by Doherty [3] and Chireix [4]. The Doherty technique emerges very favorably from this closer scrutiny. It seems to offer more than its protagonists have proposed to date, and even has claims as a linearization method in its own right. The Chireix outphasing method, on the other hand, does not seem to stand up quite so well to modern CAD analysis, but some of its elements are well worth studying.

2.2 The Doherty PA

2.2.1Introduction and Formulation

The “classical” Doherty PA (DPA) was analyzed in some detail in RFPA (Chapter 8). The configuration analyzed here, Figure 2.1, still uses two active devices but assumes a more generalized transistor transfer characteristic. So the basic elements are the two devices themselves, an impedance inverter, and a common RF load resistor. The impedance inverter can be considered conceptually to be a simple quarter-wave transmission line transformer, whose terminal characteristics have the form

although a practical implementation may beneficially use other networks to achieve this functionality. The active devices are, for the purposes of this analysis, assumed to be conducting different fundamental current amplitudes, Im and Ip, at any given input signal amplitude vin, where

Harmonic short

Figure 2.1 Schematic for two-device Doherty PA.

I m = f m (vin ), I p = f p (vin )

which are not necessarily simple linear functions of the input drive signal vin; this is a substantial generalization of the simple case considered in RFPA, where ideal linear transconductive dependencies were assumed. Ideal harmonic shorts are assumed to be placed across each device, so that only fundamental voltage and current components are considered in the analysis.

Referring to the nomenclature of Figure 2.1, (2.1) can be expanded to

give

and the remaining circuit relation is

We require expressions for the voltages at each device output, Vm and Vp, in terms of the device currents Im and Ip; clearly (2.2) gives one such relationship straight away and shows that the peaking device output voltage, which is one

and the same as the final output load voltage, is proportional to the main device current, Im; in other words it is independent of the value of Ip. Thus, provided the main device voltage is kept below clipping levels by the action of the peaking device, the linearity of the final assembly is defined entirely by the main device characteristic fm(vin). The requirement for the peaking device to perform this function can be determined from (2.3) and (2.4),

showing the possibility of using a suitable peaking device characteristic fp(vin) to “neutralize” the rapidly rising main device voltage and keep it below clipping level over the entire input signal dynamic range.

Given that this neutralization process is clearly feasible through (2.5), (2.2) emerges as a remarkable result; its significance can easily be missed in the typical idealizations used in the analysis of a standard Doherty configuration. The key point is that whatever gyrations the fp(vin) function makes in order to perform its task, and however approximate, imperfect, or nonlinear they may be, the action of the peaking device remains invisible at the output load, whose voltage remains proportional to Im.

This remarkable property verges on being a linearization process; in a typical configuration, most of the RF output power in the upper range is being supplied by the peaking device, which will most probably be a substantially nonlinear device. The dependency of the output power on the input drive signal, however, remains as defined by the main device characteristic, which can be much more linear.

This property will now be further illustrated in some specific examples. Some further nomenclature needs to be defined, in order to allow for more generalized cases. Of particular importance in any Doherty PA will be the relative Imax values for the main and peaking devices; this forms one of the essential starting parameters for a design. Throughout this analysis, the symbols Im, Ip are the amplitudes of the fundamental components of the main and peaking device currents, and not their instantaneous values. This is common enough practice in ac analysis and does not need further justification, but it does pose some hazards for the unwary; this applies especially to the definition of “Imax” values for both devices, which will again refer to the maximum values of the fundamental component of current for each device, and not to its physical “Imax” value. For this reason, the “Imax” symbolism will be avoided:

(I m )max = I M

(I p )max = I P

I P = Γ

I M

The signal drive range will, as usual, be normalized so that

0 < vin < 1

and the “breakpoint” at which peaking current starts to flow is defined to be

vin = vbk, where clearly 0 < vbk < 1.

One of the features which will be explored in this section is the possibility of using devices which have a nonlinear characteristic, rather than the “linear-dogleg” assumption which is typically used. In this respect we are taking up a similar change in mindset as was applied and discussed at some length for Class AB modes in Chapter 1 (Section 1.3). In such cases, the breakpoint is effectively replaced by the nonlinear characteristics of the device and ceases to be a major design parameter.

2.2.2 The Classical Doherty Configuration

For a “classical” Doherty configuration,

vbk = 0.5

With these ideal device characteristics, the choice of the circuit parameters R and Zo is straightforward. The peaking device is inactive up to the breakpoint, and the R and Zo values are chosen such that the main device voltage reaches its prescribed maximum allowable value at the breakpoint drive level, vin = vbk. This voltage maximum will usually correspond to the clipping level, which in turn is approximated by the dc supply; as with any PA design, some allowance will usually be included on the dc supply itself due to the knee or turn-on region of the device. There may be other considerations, given that

the DPA action enables this level to be a design parameter, rather than a physical one. For clarity, however, we will assign the symbol Vdc to this voltage limit. Thus, at the breakpoint drive level, (2.5) gives

A second relationship between the circuit parameters can be obtained by considering the maximum drive condition, vin = 1; from (2.5), the voltage amplitude at the main device is given by

the value of Zo is therefore selected such that Vm has the same value at the maximum drive as it did at the breakpoint; so

Equations (2.6) and (2.7) can be solved to give

noting that the load presented to the main device at the breakpoint is

2Vdc = 4R I M

Note that this value is a factor of two higher than would normally be used for a conventional linear design with no peaking device. This is a simple illustration of a novel way of viewing the action of a Doherty PA; the action of the peaking device is to enable the use of a much higher value of load resistor for a given main device yet still maintain similar transconductance linearity over the whole input drive range. The specific values for R and Zo are predicated in this case through the idealized device characteristics. In practice, both values will be subject to some optimization. The main objective in this

treatment of the subject is to show that useful Doherty action can be obtained for quite a wide range of real device characteristics, some of which do not even closely approach the ideal models chosen in this simplest and classical analysis.

Figure 2.2 shows the results of the above analysis, in terms of the voltages and currents at each device. The key feature is the maintenance of a constant voltage swing at the main device in the upper regime. This will result in high efficiency for the main device over the entire extent of the upper regime, and a much better efficiency/backoff characteristic than would be obtained from a conventional design. Efficiencies are not plotted at this juncture; until some more specific details are established about the implementation of the two amplifiers, it is not strictly possible to derive an efficiency curve. It is another focus of the present analysis to explore a wider range of possibilities, rather than just following the conventional assumption that the main device is an ideal Class B PA.

Any attempt to realize such a configuration runs into an immediate problem with the defined characteristics of the peaking PA. What appears to be required is a device which remains completely shut off up to the

Vin

Figure 2.2 Classical Doherty PA; current and voltage (fundamental amplitudes) characteristics for main and peaking devices plotted against input drive signal amplitude.

breakpoint, which in this case is 6 dB backed-off from the maximum drive level. Not only that, but once conduction starts, it has to make up for lost territory and race up to the IM value of the main device, with only half as much change in drive voltage. Radio frequency integrated circuit (RFIC) designers have some advantages in that they can consider a different doping or implant schedule for the peaking device. But even then, the problem of holding the device off over all but the upper few decibels of drive power remains. The most obvious and direct route to a possible solution is to use a device biased well beyond its cutoff—a Class C amplifier. There are several practical problems involved in doing this; the extent to which the device has to remain cut off requires “deep” Class C bias, which then raises breakdown issues. It also raises gain and device periphery issues. However, it is a starting point and was the original method used by Doherty, with tubes, to demonstrate this configuration.

In order to incorporate PAs with a wide range of cutoff and conduction angles, it is necessary to recall some of the results on the basic analysis of these modes. Figure 2.3 shows a plot of the fundamental component of current versus sinusoidal drive voltage for various quiescent bias settings for a device having the conventional ideal “linear-dogleg” characteristic defined in Chapter 1. At first sight, the Class C (Vq < 0) curves seem to show useful possibilities for realizing the peaking device characteristics, assuming that the main device followed something close to the Class B (Vq = 0) curve. A quiescent bias, for example, of Vq = −0.5 would result in something very close to the required “holdoff” range shown in Figure 2.2. But if both devices are

Figure 2.3 Fundamental current component for reduced conduction angle mode device; device has linear conduction range 0 < Vin < 1; quiescent bias settings (Vq) measured in same units as Vin.

matter of extending the reduced conduction angle analysis shown in Figure 2.3 to include the dc components. The result is essentially the classical Doherty efficiency curve for a symmetrical configuration and a 6-dB breakpoint. The slightly sluggish start to the peaking PA conduction, caused by the Class C operation, causes some variation in the main PA voltage beyond the breakpoint. This also necessitates reducing the Zo value slightly in order to keep the main PA voltage swing below the stipulated supply rail value. The key feature is the return to Class B efficiency at the breakpoint, where the peaking device shuts down; thereafter as the drive power is further reduced, the efficiency drops down the conventional curve for a Class B PA.

2.2.3Variations on the Classical Configuration

As an introduction to more generalized, and practical, implementations of the Doherty PA, Figure 2.5 shows the use of a nonlinear device as the peaking PA. This is picking up a theme from Chapter 1 (see Section 1.3) where the concept of using a nonlinear characteristic, rather than using the cutoff of a linear device, was explored for Class AB applications. In order to illustrate an important point, a somewhat arbitrary device characteristic is shown. The two device characteristics are plotted in Figure 2.5; clearly, there is a distinct “violation” of the original requirements for the peaking device, in that the peaking device never fully shuts down, and is palpably nonlinear. The voltage curves, therefore, tell an interesting story. First and foremost, the peaking device voltage Vp still shows the ideal linear response pertinent to Class B operation. The main device voltage shows only an approximation to the required characteristic. The less abrupt main device voltage “cornering” will have some impact on the efficiency, as shown in Figure 2.5. As always in DPA analysis, the efficiency curve does not follow implicitly from the voltage and current relationships and depends on the more detailed implementation of the peaking device characteristic; in this case an ideal cube-law device at zero quiescent bias has been assumed for the peaking device.

Figure 2.6 shows the efficiency backoff plot for three cases: ideal Doherty using a Class C peaker with suitable periphery and bias offset adjustment, a simple Class B PA, and the less-ideal peaker realization shown in Figure 2.5. It will be seen that the efficiency impact for the nonlinear peaker is significant but not by any means fatal. But the other issue is to recognize that the nonlinearity of the peaking PA does not show up in the final transfer characteristic; the function of the peaker is to pull the voltage at the main PA lower than the maximum value stipulated by the rail supply. Indeed, once the basic principle that linearity is maintained regardless of the

Figure 2.8 shows such a realization, which now allows the efficiency to be computed. Compared to the ideal characteristics shown in Figure 2.7, some adjustment has to be made to the Zo value in order to keep the main device voltage below the allowable maximum value. The efficiency curve still looks quite attractive, despite an inevitably deeper dip in the mid-backoff region. Such an asymmetrical DPA would, on most counts, appear to be a more worthwhile target than the classical symmetrical version, for practical realization in modern applications.

The asymmetrical DPA would normally be associated with high peak- to-average power ratio signals, such as occur in wideband CDMA (WCDMA) or multicarrier systems. In such applications, the use of a peaking device with several, or many, times the peak current of the main device can greatly improve overall efficiency. There is, however, another side to the asymmetrical coin; there may be some benefit in having a peaking device with a lower peak current. One of the reasons for examining this concept is that in practice this will be the case if two similar devices are used, in an attempt to realize a classical configuration. Looking again at Figure 2.3, if the main device is run in Class B (Vq = 0), and a similar device is biased into Class C (say, at Vq = −0.5), it is clear that at the drive level required to obtain

Vin

Figure 2.8 Asymmetrical Doherty PA; peaking function realized using Class C and scaled periphery device.

the maximum peak current in the main device, the peaking device is only producing a fundamental component of about 40% of the main device component, IM. The result of this is that the peaking device current will always be insufficient to pull down the main device voltage, and therefore a lower load resistance value must be presented to it, by reducing the Zo value. This situation is shown in Figure 2.9, and is here termed Doherty-Lite.

The key point about the Doherty-Lite configuration is that a significant improvement in backoff efficiency can be obtained with the simplest possible circuit configuration. The efficiency plot will not, however, have the classical “twin peaks.” We will see in Section 2.2.4 that there is a reality to be faced with any of the “heavier” Doherty PAs described thus far, and in other literature. Practical realization of the peaking PA function by scaling of the device periphery and Class C biasing runs into an escalation of practical difficulties. Alternatives, in the form of bias adaption and DSP amplitude control, need to be pursued. But the Doherty-Lite approach gives some more modest benefits whilst retaining a viable and simple practical circuit. Figure 2.10 shows the peaking current and efficiency characteristics for three bias settings of the peaking device. The benefits of Doherty-Lite operation seem

Vin

Figure 2.9 “Doherty-Lite” PA using same device type for main and peaking functions. (Unlabeled dotted line is conventional Class B efficiency, for comparison. Normalized circuit element values: R = 1, Zo = 1.625.)

Vin

Figure 2.10 “Doherty-Lite” PA; various bias settings for peaking device: Vpq = −0.5, −0.3, and −0.1. Zo is adjusted in each case. Dotted line shows conventional Class B efficiency characteristic.

to have a broad optimum in the vicinity of vpq = −0.3; this gives less efficiency improvement in the upper drive range, but quite substantial improvements in the −10-dB backoff region, in comparison to a conventional Class B arrangement. The power contribution of the peaking device will also be greater as the breakpoint is set to lower PBO values.

At this point, we seem to have a configuration which is sufficiently viable in a practical sense, to run a full scale simulation using “real” device models. Figure 2.11 shows such a simulation, where the main and peaking devices are identical, but the main device has been optimized for a reasonable efficiency/linearity compromise. The peaking device is biased so that it becomes active at about the −10-dB backoff point. There is clearly a major efficiency improvement in the Doherty-Lite case, amounting to about 20% in the midrange, and 10% over most of the 20-dB range. The peaking device adds about 2 dB, rather than the full 3 dB which would be expected from two equal periphery devices. There is also a small linearity degradation evident on the Doherty plot. This can be traded with efficiency by fine-tuning the RF load and Zo values. This simulation, which includes detailed models